1. Introduction

The phenomenon dealing with extraordinary transmission [1] and directional beaming of light [2] has stimulated much interest over the last years. Based on the surface plasmon polariton (SPP) propagation on the metallic surface, a series of novel applications such as nanofocusing [3–5], nanoguiding of light [6,7] has been demonstrated. A metallic nanoslit flanked by periodic metallic corrugations [2,8–10], dielectric gratings [11,12], and surrounded by metal heterostructures [13] can be used for enhanced transmission or directional beaming of light. Huygens’s principle [4,9,10], rigorous coupled wave theory [8] and guided mode resonance theory [14] are used to design the structures and interpret the physics behind the extraordinary transmission as well as directional beaming phenomenon. Recently, dual-wavelength enhanced transmission of light is realized by a periodic metallic corrugations flanked metal nanoslit, of which two wavelengths are enhanced by the left and right corrugations separately [15], but this method is not proper for dual-wavelength directional beaming because of the cross effect of the left and right corrugations. We previously proposed a dielectric gratings flanked metal nanoslit for directional beaming of dual-wavelength light [14], which is particularly interesting in optical data storage (one for writing and the other for reading). However, the transmission efficiency of such beaming light is limited. In this paper, we go on designing a single nanoslit in an optically thick metal film flanked with dielectric gratings in both input and output surfaces for enhanced dual-wavelength light beaming.

2. Theoretical design

Figure 1 illustrates the proposed structure, where a nanoslit with a width of w = 100 nm is introduced in the center of a metal Ag film with a thickness of h_m = 200 nm. The input surface is covered by a dielectric grating with thickness h_i and period Λ_i = a_i + b_i, while the output surface is covered by the other one with thickness h_o and period Λ_o = a_o + b_o, where a_i (a_o) and b_i (b_o) are the widths of the dielectric films (with dielectric constant ε_d) and air gaps (ε₀) of the input (output) gratings, respectively. When a TM-polarized (magnetic field parallel to the y direction) plane electromagnetic wave with wavelength λ is incident from the input grating at an angle θ, it couples into surface modes (SPPs) and propagation modes on the input metallic surface firstly, and then into the SPPs in the metal nanoslit and surface modes and propagation modes on the output metallic surface, and finally couples out to propagation modes in the air by the diffraction of the output dielectric grating at an angle θ' [13,14,16].

 

Fig. 1 (Color online) Proposed structure for transmission enhanced directional beaming of dual-wavelength light. For parameters in the figure, see the text.

Download Full Size | PDF

The input and output gratings can be designed separately, since the input and output gratings are responsible for the transmission enhancement and directional beaming of light independently [2,9,10,12]. If normal incidence (θ = 0) and directional beaming [θ' = 0, Fig. 1] are assumed, the coupling in and coupling out are inverse processes, which means the input and output gratings share the same structure parameters (Λ_i = Λ_o, a_i = a_o, b_i = b_o, h_i = h_o). The coupling between propagation modes in the air at an angle of ϕ (stands for θ or θ') and the modes on the input and output metallic surface for two different wavelengths λ₁ and λ₂ is governed by the grating Eq. (8),14]

If the filling factor ƒ = 0.93 and the dielectric constants of the Ag film and the dielectric layer (Si₃N₄) are used as ε_m1 = −6.6086 + i0.1947, ε_d1 = 4.2243 at λ₁ (442 nm) and ε_m2 = −18.3132 + i0.4981 and ε_d2 = 4.0828 at λ₂ (633 nm), respectively [19,20], a solution of Eq. (2) at h_i = 192 nm is obtained as β_rλ1 = β_rλ2 = β_r = 2.169 × 107 m⁻¹ with TM₁(λ₁) and TM₀(λ₂) modes in the structure. TM_m(λ_n) denotes the m-th order TM mode for wavelengths λ_n (n = 1, 2). Then we get grating period Λ_i = 2π/β_r = 290 nm from Eq. (1) for θ = θ' = 0 with a_i = 270 nm and b_i = 20 nm. The above analysis provides the theoretically predicted structure parameters (TPSPs: a_i = a_o = 270 nm, b_i = b_o = 20 nm, h_i = h_o = 192 nm) for enhanced directional beaming of dual-wavelength light simultaneously.

3. Numerical simulation and analysis

In the following, we use finite-difference time-domain (FDTD) numerical simulation to demonstrate the above prediction. The number of grating period is set to be 19 for both input and output gratings while optimized structure parameters for the output gratings (OSPs: a_o = 275 nm, b_o = 20 nm and h_o = 195 nm) are slightly different from the TPSPs determined by the previous method [14]. For the input gratings, the OSPs (a_i = 255 nm, b_i = 20 nm and h_i = 220 nm) are determined by the method which we will discuss below.

Figure 2(a) illustrates the FDTD simulated dependence of normalized-to-area transmittance (defined as transmittance normalized to the incident flux in the direction normal to the metal surfaces and to the slit area) on wavelength for different structures. For a structure only with output gratings (dash-dotted black curve), we see that these is no transmission peak, confirming that the output surface shows little effect on the transmittance compared to the input surface [9,10,12]. For a structure with TPSPs at input and output gratings (dashed blue curve), there are two transmission peaks at 439 nm (Peak 1) and 650 nm (Peak 2) appeared. For the structure with OSPs at input and output gratings (solid red curve), peaks 1 and 2 shift to 442 nm and 633 nm, respectively. Figures 2(b) and 2(c) show the FDTD simulated wavelength dependence of peak 1 around 442 nm (Lines L1, L3) and peak 2 around 633 nm (Lines L2, L4) on the input grating thickness h_i [Fig. 2(b)] and period Λ_i [Fig. 2(c)] while the other parameters are at OSPs (a_i = 255 nm, b_i = 20 nm, a_o = 275 nm, b_o = 20 nm and h_o = 195 nm) and (b_i = 20 nm, h_i = 220 nm, a_o = 275 nm, b_o = 20 nm and h_o = 195 nm), respectively. The slopes of Lines L1, L2, L3 and L4 shown in Figs. 2(b) and 2(c) are 0.36, 0.11, 0.65 and 1.57. Since peak 1 is sensitive to both h_i and Λ_i (see Lines L1 and L3), but peak 2 only sensitive to Λ_i (Lines L2 and L4), we shift peak 2 from 650 to 633 nm by reducing Λ_i and then shift peak 1 to 442 nm by increasing h_i in TPSPs [see Fig. 2(a)]. Figure 2(d) shows the transmittance of the beaming light on the error δ of grating structure parameters a_i and h_i [see Fig. 1] for λ₁ = 442 nm (Curve C1 for a_i and Curve C3 for h_i) and λ₂ = 633 nm (Curve C2 for a_i and Curve C4 for h_i), where a_i and h_i corresponds to the designed 255 nm and 220 nm, respectively. From Fig. 2(d) we see that the peak transmittance shows a reduction with the increased error δ. This is due to the broken of the transmission enhancement condition Eq. (1) by deviation of the grating period Λ_i( = a_i + b_i) (Curve C1 and Curve C2) or β_r determined by the grating thickness h_i (Curve C3 and Curve C4). The transmittance is sensitive to a_i (Curve C1 and Curve C2) as the transmission enhancement condition Eq. (1) is sensitive to Λ_i( = a_i + b_i). Since β_r of the surface mode TM₀(λ₂) is not sensitive to h_i when h_i >150 nm [13,14], the transmittance changes slower for TM₀(λ₂) around h_i (Curve C4) than TM₀(λ₂) around a_i (Curve C2). From the above analysis we can see that the OSPs for input gratings (a_i = 255 nm, b_i = 20 nm, h_i = 220 nm) could be obtained by fine adjustment of TPSPs, at which normalized-to-area transmissions of 10.8 for λ₁ and 14.7 for λ₂ are achieved.

 

Fig. 2 (Color online) (a) FDTD simulated dependence of normalized-to-area transmittance on wavelength for the structure with only output grating (dash-dotted black curve), TPSPs (dashed blue curve) and OSPs (solid red curve), respectively. (b), (c) Wavelength dependence of peak 1 around 442 nm (Lines L1, L3) and peak 2 around 633 nm (Lines L2, L4) on the input grating thickness h_i [(b)] and period Λ_i [(c)], respectively, while the other parameters are at OSPs. (d) Normalized-to-area transmittance of the beaming light on the error δ of grating structure parameters a_i and h_i for λ₁ = 442 nm (Curve C1 for a_i and Curve C3 for h_i) and λ₂ = 633 nm (Curve C2 for a_i and Curve C4 for h_i), respectively, while the other parameters are at OSPs.

Download Full Size | PDF

Figure 3(a) presents the FDTD simulated dependence of HAHMs (half angles at half-maximum) and DEs (defined as the total energy confined in the full width at half maximum of the central beam divided by light energy radiated into the free space) of emitted central beams on wavelength deviation Δλ from λ₁ = 442 nm (Curve C5 and Curve C7) and λ₂ = 633 nm (Curve C6 and Curve C8) with OSPs at both input and output grating, respectively. The data are obtained along the x axis on a line z = 60 μm away from the output gratings. From Curve C5 we see that HAHM of the beaming light around λ₁ decreases to about 3° as Δλ is increased from −12 nm to −4 nm and keeps steady as Δλ is increasing to 10 nm, and then rise up as Δλ is further increased, while Curve C7 shows the opposite properties. This can be understood by calculating the diffraction angle θ' of the emitted light through submitting β_r of TM₁ [inset of Fig. 3(a)] into Eq. (1). The θ' decreases from positive [β_r>2.169 × 107 m⁻¹, spreading] to zero [β_r = 2.169 × 107 m⁻¹, directional beaming] and then to negative [β_r<2.169 × 107 m⁻¹, focusing] as Δλ changes from −12 nm to 12 nm. The two beams emitted from the left and right half of the output gratings overlapped [14], which results in a relatively flat range of Curve C5 and Curve C7, as Δλ increasing from −4 nm to 5 nm. For wavelength around λ₂, the slope of the dispersion curve of TM₀ is smaller than that of TM₁ [Fig. 3(a), inset], which would result in a relatively flat range of Curve C6 and Curve C8 in an even larger range of Δλ from −12 nm to 12 nm. Note from Fig. 3(a) that the HAHMs are 2.40° and 2.65°, while the DEs are 0.43 and 0.39 at λ₁ and λ₂ respectively, implying that the enhanced directional beaming of dual-wavelength light λ₁ and λ₂ are achieved simultaneously.

 

Fig. 3 (Color online) (a) FDTD simulated dependence of HAHMs and DEs of emitted central beams on wavelength deviation Δλ from λ₁ = 442 nm (Curve C5 and Curve C7) and λ₂ = 633 nm (Curve C6 and Curve C8). Inset, dispersion relation of TM₁ (solid blue line) and TM₀ (dotted red line) from Eq. (2) with h_i = 220 nm. (b) Dependence of the normalized transmittance and HAHMs of the emitted central beams on the incident angle |θ| for λ₁ = 442 nm (Curve C9 and Curve C11) and λ₂ = 633 nm (Curve C10 and Curve C12). OSPs are used at both input and output gratings.

Download Full Size | PDF

Figure 3(b) shows the dependence of the normalized transmittance and HAHMs of emitted central beams on the incident angle |θ| for λ₁ = 442 nm (Curve C9 and Curve C11) and λ₂ = 633 nm (Curve C10 and Curve C12) with OSPs at both input and output gratings, respectively. From Curve C9 (C10) we see that the normalized transmittance of the beaming light decreases to about 1 (3) for λ₁ (λ₂) as |θ| is increased from 0° to 5° (7°) and keeps steady as |θ| is increasing to 20°, while Curve C11 (C12) keeps steady as |θ| is increased from 0° to 5° (7°) and then rise up rapidly. The grating equation Eq. (1) is broken as |θ| is increased for the structure with OSPs at the input grating, which results in a decreasing coupling efficiency from the incident light to SPPs on the input metallic surface, and then the decreasing transmittance (Curve C9 and Curve C10). With the increase of |θ|, the near field distribution of light in the central slit is changed, which results in a field redistribution in the far field and then a broadening of the HAHMs. From the above analysis we can see that the enhanced directional beaming of dual-wavelength light λ₁ and λ₂ can be realized with |θ|≤5°, respectively.

4. Conclusion

In summary, we have designed a dielectric-metal-dielectric structure to realize the transmission enhanced dual-wavelength light beaming based on guided-mode resonance theory. A single nanoslit in an optically thick metal film flanked by dielectric gratings on both input and output surfaces of the slit can realize transmission enhanced directional beaming of two wavelength lights (442 nm and 633 nm) simultaneously. FDTD simulations well demonstrate the theoretical prediction. The proposed structures would be interesting in high density optical data storage, spatial and spectral multiplexing, nanoscale directional light sources and emitters, biosensors, near-field optics and spectroscopy, etc.